DOCSLIB.ORG

Embedding Finite Sets in a Logic Programming Language

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Embedding Finite Sets in a Logic Programming Language Embedding Finite Sets in a Logic Programming Language Agostino Dovier Eugenio G. Omodeo Dipartimento di Informatica Dip. di Informatica e Sistemistica Università di Pisa Università di Roma La Sapienza C.so Italia 40, 56100 PISA Via Salaria 113, 00198 ROMA [email protected] [email protected] Enrico Pontelli Gianfranco Rossi New Mexico State Univ. Dip. di Matematica Dept. of Computer Science Università di Bologna Las Cruces, NM 88003 P.zza di Porta S.Donato 5, [email protected] 40127 BOLOGNA [email protected] Abstract. A way of introducing simple (finite) set designations and operations as first-class objects of an (unrestricted) logic programming language is discussed from both the declarative and the operational semantics viewpoint. First, special set terms are added to definite Horn clause logic and an extended Herbrand Universe based on an axiomatic characterization of the kind of sets we are dealing with is defined accordingly. Moreover, distinguished predicates representing set membership and equality are added to the base language along with their negative counterparts (π and ). A new unification algorithm which can cope with set terms is developed and proved to terminate. Usual SLD resolution is modified so as to incorporate the new unification algorithm and to properly manage the distinguished predicates for set operations (in particular, conjunctions of atoms containing π and are dealt with as constraints, first reduced to a canonical form through a suitable canonization algorithm). Finally, the application of the resulting language to the definition of Restricted Universal Quantifiers is discussed. 1 Introduction General agreement exists about the usefulness of sets as very high-level representations of complex data structures (cf. [2]) and of set abstractions in code specifications. In particular, sets and set abstractions can be conveniently used in rapid software prototyping, where efficiency is not a primary requirement while the availability of high level data and operation abstractions are key features of the prototype implementation language. Only relatively few programming languages embody sets as primitive objects. Among them, the (non-executable) specification language Z [26], the procedural language SETL [5, 24] and the functional languages MIRANDA [25] and ME TOO [18]. A number of recent proposals have preferred, however, a more declarative programming framework. In particular, logic programming languages have been advocated by many as a suitable setting for declarative programming with sets. Among them, the proposals of the logic database languages LDL [3] and LPS [11]. Also the recent work by Legeard and Legros [13] introduces sets in a Prolog-like language dealing with set expressions as constraints, while [23] analyzes the problem of logic programming with sets in its generality (including infinite sets). A similar framework but stressing the notion of equational programming is also advocated in [9]. Actual Prolog systems already provide some facilities to support sets in the form of the built-in predicates setof and bagof. However, it is widely recognized that the definition of these facilities is quite unsatisfactory. In fact, no precise logical semantics can be attributed to them. Sets are simply represented as lists and therefore are dealt with as ordinary terms (e.g. by an implicit order among their elements). On the contrary, a logic language embodying sets ought to supply special set former terms along with a few basic operations on them as part of the language, and allow other more complex operations on sets (possibly including the setof operation), to become definable within the language itself, without having to further extend its semantics. Aim of this paper is to define such an extended logic language, called {log} (read set-log). Unlike some of the above cited proposals, however, we do not restrict ourselves to any specific application domain. So our starting point is a pure logic programming language, that is Definite Horn clauses with no extra-logical constructs. Then simple set constructs are added to this language (namely, enumerated set terms, set membership and equality and their negative counterparts) and suitable operational and declarative semantics are defined for them. It is shown that the usual set operations, such as union, intersection, etc., can be defined in this extended language. Also Restricted Universal Quantifiers are shown to be definable within the language itself; so they can be added to it as a simple syntactic extension (dealt with by a suitable preprocessor). The paper is organized as follows. Section 2 briefly describes a few extensions to ordinary logic programming language syntax that can support the introduction of set terms. Declarative semantics of the resulting language is presented in section 3 by defining an extended Herbrand Universe based on an axiomatic characterization of the kind of sets we are dealing with. The problem of unification of set terms is then addressed in section 4 and a new unification algorithm which can deal with set terms is described in detail. Section 5 presents the extended SLD procedure incorporating the new unification algorithm and the management of , =, and π, based on the so called constraint canonization algorithm. Section 6 shows how Restricted Universal Quantifiers can be defined in our language. Finally, a comparison with some other related proposals is carried out in section 7. 2 {log} syntax To begin with, we introduce in this section the extensional representation of finite sets. All that is presupposed is the availability of: - an interpreted constant, {}, for the empty set Ø; - a binary function symbol, with (used as an infix left associative operator), to be interpreted as follows: s with t stands for the set that results from adding t as a new element to the set s. Apart from these two symbols, {log} contains the usual equipment of clausal Horn logic (cf. [14]) along with distinguished predicates for set membership and equality ( and =) and their negative counterparts (π and ). The extensional representation for sets referred to above is provided by a collection G of ground terms: this is the smallest collection such that - the constant {} belongs to G ; - s with t, s, t ground terms (not necessarily in G), belongs to G . In view of the intended interpretation, any s in G will be called a set term. A non-ground term t is called a set term if there exists an instantiation σ of the variables in t such that tσ belongs to G; in particular a variable is a set term. Set terms of the form t with tn with ... with t1 wherein t is not a set term are intended to designate sets based on a kernel t other than {} (also called colored sets). For the sake of simplicity we introduce special syntactic forms to designate set terms: {t1,...,tn|s} stands for s with tn with ... with t1 and {t1,...,tn} stands for {} with tn with ... with t1 where n ≥ 1 and s, t1,...,tn are terms. For example: - f(a,{5}), i.e. f(a,{} with 5), is a term, but not a set term; - {2,g(3),a}, i.e. {} with a with g(3) with 2, is a ground set term; - {}, {1,X,Y,2}, {1,1,{2,{}},f(a,{b})} and any term {t1,...,tn|R} with a 'tail' variable R, are set terms - {a|f({b})} is a colored set term based on the kernel f({b}). For the rest of this paper, we will freely exploit the syntactic features available in Edinburgh Prolog in addition to the constructs discussed above. Three sample {log} clauses are: - q(X) :- X {a,b,c,d} & p(X) - singleton(X) :- X = {Y}. - in_difference(X,Set1,Set2) :- X Set1 & X Set2. Colored set terms (i.e. terms of the form {t1,...,tn|t} wherein t is not a set term) do not designate sets of any conventional kind; one might therefore contend that such terms ought to be forbidden as they are unlikely to serve any serious purpose in programming. Nevertheless, we deem it convenient to always regard {t1,...,tn|t} as a legal set term when t1,...,tn, t are legal, to make the language structure absolutely uniform and the inference mechanisms (e.g. unification) more straightforward. 3 {log} declarative semantics To convey the meaning of our language, we are to formally characterize the legal interpretations of {log}. We begin by providing the axioms of a suitable first-order set theory with equality: the legal interpretations of {log} will be the models of these axioms (cf. [8]). Next we will focus on a privileged interpretation domain formed by terms, UH, resulting from suitable modifications to the classical Herbrand universe [14]; moreover, we will designate a fixed binary relation over UH as the privileged interpretation of . (The predicate = will be interpreted as syntactic equality). The interpretation of the predicate symbols of {log} other than or = is left totally unspecified for the time being, as it must take into account a {log} program, viz. a set of application-specific clausal axioms enriching the initial axiom endowment we are presenting now. 3.1 Set axioms Here are our basic axioms (X, Y, V, S and the Xis distinct variables ranging over the whole interpretation domain): (Z) V {}; (W1) V π Y ∅ (V X with Y × V X); (W2) Y X with Y; (L) Y X ∅ ∃z (Y z & X = z with Y); (K0) (∀z z X) ∅ (X = ker(X)); (K1) V ker(X); (K2) ker(V with Y) = ker(V); (E) (ker(X) = ker(Y) & ∀z (z X × z Y)) ∅ X = Y; (R) ∃z ∀w (w X ∅ (z X & w z)); (U) f(X1,...,Xn) π {} & V f(X1,...,Xn), where n is the arity of f and f/n {{}/0, ker/1, with/2}. Although rather conventional in the overall, and somewhat narrow in comparison to well-established set theories such as Zermelo-Fraenkel or von Neumann-Bernays-Gödel, our theory slightly deviates from the classical ones under two respects: - Presence of urelements.
Load more

Recommended publications
  • The Development of Mathematical Logic from Russell to Tarski: 1900–1935
    The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu Richard Zach Calixto Badesa The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu (University of California, Berkeley) Richard Zach (University of Calgary) Calixto Badesa (Universitat de Barcelona) Final Draft—May 2004 To appear in: Leila Haaparanta, ed., The Development of Modern Logic. New York and Oxford: Oxford University Press, 2004 Contents Contents i Introduction 1 1 Itinerary I: Metatheoretical Properties of Axiomatic Systems 3 1.1 Introduction . 3 1.2 Peano’s school on the logical structure of theories . 4 1.3 Hilbert on axiomatization . 8 1.4 Completeness and categoricity in the work of Veblen and Huntington . 10 1.5 Truth in a structure . 12 2 Itinerary II: Bertrand Russell’s Mathematical Logic 15 2.1 From the Paris congress to the Principles of Mathematics 1900–1903 . 15 2.2 Russell and Poincar´e on predicativity . 19 2.3 On Denoting . 21 2.4 Russell’s ramified type theory . 22 2.5 The logic of Principia ......................... 25 2.6 Further developments . 26 3 Itinerary III: Zermelo’s Axiomatization of Set Theory and Re- lated Foundational Issues 29 3.1 The debate on the axiom of choice . 29 3.2 Zermelo’s axiomatization of set theory . 32 3.3 The discussion on the notion of “definit” . 35 3.4 Metatheoretical studies of Zermelo’s axiomatization . 38 4 Itinerary IV: The Theory of Relatives and Lowenheim’s¨ Theorem 41 4.1 Theory of relatives and model theory . 41 4.2 The logic of relatives .
    [Show full text]
  • A Functional Hitchhiker's Guide to Hereditarily Finite Sets, Ackermann
    A Functional Hitchhiker’s Guide to Hereditarily Finite Sets, Ackermann Encodings and Pairing Functions – unpublished draft – Paul Tarau Department of Computer Science and Engineering University of North Texas [email protected] Abstract interest in various fields, from representing structured data The paper is organized as a self-contained literate Haskell in databases (Leontjev and Sazonov 2000) to reasoning with program that implements elements of an executable fi- sets and set constraints in a Logic Programming framework nite set theory with focus on combinatorial generation and (Dovier et al. 2000; Piazza and Policriti 2004; Dovier et al. arithmetic encodings. The code, tested under GHC 6.6.1, 2001). is available at http://logic.csci.unt.edu/tarau/ The Universe of Hereditarily Finite Sets is built from the research/2008/fSET.zip. empty set (or a set of Urelements) by successively applying We introduce ranking and unranking functions generaliz- powerset and set union operations. A surprising bijection, ing Ackermann’s encoding to the universe of Hereditarily Fi- discovered by Wilhelm Ackermann in 1937 (Ackermann nite Sets with Urelements. Then we build a lazy enumerator 1937; Abian and Lamacchia 1978; Kaye and Wong 2007) for Hereditarily Finite Sets with Urelements that matches the from Hereditarily Finite Sets to Natural Numbers, was the unranking function provided by the inverse of Ackermann’s original trigger for our work on building in a mathematically encoding and we describe functors between them resulting elegant programming language, a concise and executable in arithmetic encodings for powersets, hypergraphs, ordinals hereditarily finite set theory. The arbitrary size of the data and choice functions.
    [Show full text]
  • A Theory of Particular Sets, As It Uses Open Formulas Rather Than Sequents
    A Theory of Particular Sets Paul Blain Levy, University of Birmingham June 14, 2019 Abstract ZFC has sentences that quantify over all sets or all ordinals, without restriction. Some have argued that sen- tences of this kind lack a determinate meaning. We propose a set theory called TOPS, using Natural Deduction, that avoids this problem by speaking only about particular sets. 1 Introduction ZFC has long been established as a set-theoretic foundation of mathematics, but concerns about its meaningfulness have often been raised. Specifically, its use of unrestricted quantifiers seems to presuppose an absolute totality of sets. The purpose of this paper is to present a new set theory called TOPS that addresses this concern by speaking only about particular sets. Though weaker than ZFC, it is adequate for a large amount of mathematics. To explain the need for a new theory, we begin in Section 2 by surveying some basic mathematical beliefs and conceptions. Section 3 presents the language of TOPS, and Section 4 the theory itself, using Natural Deduction. Section 5 adapts the theory to allow theorems with free variables. Related work is discussed in Section 6. While TOPS is by no means the first set theory to use restricted quantifiers in order to avoid assuming a totality of sets, it is the first to do so using Natural Deduction, which turns out to be very helpful. This is apparent when TOPS is compared to a previously studied system [26] that proves essentially the same sentences but includes an inference rule that (we argue) cannot be considered truth-preserving.
    [Show full text]
  • Sets As Graphs
    Universita` degli Studi di Udine Dipartimento di Matematica e Informatica Dottorato di Ricerca in Informatica XXIV Ciclo - Anno Accademico 2011-2012 Ph.D. Thesis Sets as Graphs Candidate: Supervisors: Alexandru Ioan Tomescu Alberto Policriti Eugenio G. Omodeo December 2011 Tesi di dottorato di Alexandru Ioan Tomescu, discussa presso l'Universit`adegli Studi di Udine Author's e-mail: [email protected] Author's address: Dipartimento di Matematica e Informatica Universit`adegli Studi di Udine Via delle Scienze, 206 33100 Udine Italia Tesi di dottorato di Alexandru Ioan Tomescu, discussa presso l'Universit`adegli Studi di Udine To my parents Tesi di dottorato di Alexandru Ioan Tomescu, discussa presso l'Universit`adegli Studi di Udine Tesi di dottorato di Alexandru Ioan Tomescu, discussa presso l'Universit`adegli Studi di Udine Abstract In set theory and formal logic, a set is generally an object containing nothing but other sets as elements. Not only sets enable uniformity in the formalization of the whole of mathematics, but their ease-of-use and conciseness are employed to represent information in some computer languages. Given the intrinsic nesting property of sets, it is natural to represent them as directed graphs: vertices will stand for sets, while the arc relation will mimic the membership relation. This switch of perspective is important: from a computational point of view, this led to many decidability results, while from a logical point of view, this allowed for natural extensions of the concept of set, such as that of hyperset. Interpreting a set as a directed graph gives rise to many combinatorial, structural and computational questions, having as unifying goal that of a transfer of results and techniques across the two areas.
    [Show full text]
  • Download (15Mb)
    University of Warwick institutional repository: http://go.warwick.ac.uk/wrap A Thesis Submitted for the Degree of PhD at the University of Warwick http://go.warwick.ac.uk/wrap/46969 This thesis is made available online and is protected by original copyright. Please scroll down to view the document itself. Please refer to the repository record for this item for information to help you to cite it. Our policy information is available from the repository home page. Supporting the Migration from Construal to Program: Rethinking Software Development by Nicolas William Pope Thesis Submitted to the University of Warwick for the degree of Doctor of Philosophy Department of Computer Science August 2011 Contents List of Tables vii List of Figures viii Acknowledgments xvi Declarations xvii Abstract xviii Abbreviations xix Chapter 1 Introduction 1 1.1 Plastic Applications . .1 1.2 Plastic Software Environments . .4 1.3 A Lack of Plasticity . .6 1.4 Thesis Aims . .9 1.5 Thesis Outline . 10 Chapter 2 Background 13 2.1 End-User Development . 13 2.1.1 Principles . 14 2.1.2 Common Approaches . 17 ii 2.1.3 Existing Environments . 17 2.1.4 Guidelines . 24 2.2 Empirical Modelling . 26 2.2.1 What is Empirical Modelling? . 26 2.2.2 The Principles . 32 2.2.3 Current Tools . 35 2.2.4 An Example Model . 38 2.2.5 EM and Software Development . 41 2.3 Miscellaneous Technologies . 43 Chapter 3 Enabling Plastic Applications 45 3.1 Empirical Modelling and Plastic Applications . 45 3.2 Dimensions of Refinement . 48 3.3 Limitations of EM Tools and Concepts .
    [Show full text]
  • Literaturverzeichnis
    Literaturverzeichnis [1] Abian, A.: Nonstandard models for arithmetic and [14] Boole, G.; Corcoran, J.: The Laws of Thought analysis. In: Studia Logica 33 (1974), Nr. 1, S. 11–22 (Reprint). New York: Prometheus Books, 2003 [2] Ackermann, W.: Zum Hilbert’schen Aufbau der [15] Boolos, G.: A New Proof of the Gödel Incompleteness reellen Zahlen. In: Mathematische Annalen 99 (1928), Theorem. In: Notices of the American Mathematical S. 118–133 Society (1989), Nr. 36, S. 388–390 [3] Ackermann, W.: Zur Axiomatik der Mengenlehre. In: [16] Boolos, G. S.; Burgess, J. P.; Jeffrey, R. C.: Mathematische Annalen 131 (1956), August, Nr. 4, S. Computability and Logic. Cambridge: Cambridge 336–345 University Press, 2007 [4] Aliprand, J.: The Unicode Standard Version 5.0. [17] Boolos, G. S.; Jeffrey, R. C.: Computability and Boston, MA: Addison-Wesley, 2007 Logic. Cambridge: Cambridge University Press, 1989 [5] Amos, M.: Theoretical and Experimental DNA [18] The Busy Beaver Game and the Meaning of Life. Computation. Berlin, Heidelberg, New York: In: Brady, A. H.: The Universal Turing Machine: A Springer-Verlag, 2005 Half Century Survey. Oxford: Oxford University [6] Baez, J.: This Week’s Finds in Mathematical Physics. Press, 1991, S. 259–277 http://math.ucr.edu/home/baez/ [19] Brady, G.: From Peirce to Skolem: A Neglected week236.html Chapter in the History of Logic. Amsterdam: Elsevier [7] Bauer, Andrej: Portraitphoto von Dana Scott. Scienceg, 2000 (Studies in the History and Philosophy http://creativecommons.org/licenses/ of Mathematics) by-sa/2.5/. – Creative Commons License 2.5, [20] Burali-Forti, C.: Una questione sui numeri transfiniti.
    [Show full text]
  • An Anthology of Philosophical Studies Volume 9
    AN ANTHOLOGY OF PHILOSOPHICAL STUDIES VOLUME 9 Editor Patricia Hanna University of Utah USA Editorial Board Maria Adamos Georgia Southern University USA Gary Fuller Central Michigan University USA Donald Poochigian University of North Dakota USA Andrew Ward University of York UK Joe Naimo University of Notre Dame Australia Australia Board of Reviewers Daniel Considine Metropolitan State College in Denver, Colorado USA. Katherine Cooklin Slippery Rock University of Pennsylvania USA Chrysoula Gitsoulis City College of New York USA Keith Green East Tennessee State University USA Dimitria Electra Gratzia University of Akron USA Philip Matthews University of Notre Dame Australia Australia Michael Matthis Lamar University USA Mark McEvoy Hofstra University USA Chris Onof Birkbeck College UK John Thompson Christopher Newport University USA Kiriake Xerohemona Florida International University USA AN ANTHOLOGY OF PHILOSOPHICAL STUDIES VOLUME 9 Edited by Patricia Hanna Athens Institute for Education and Research 2015 AN ANTHOLOGY OF PHILOSOPHICAL STUDIES VOLUME 9 First Published in Athens, Greece by the Athens Institute for Education and Research. ISBN: 978-618-5065-95-9 All rights reserved. No part of this publication may be reproduced, stored, retrieved system, or transmitted, in any form or by any means, without the written permission of the publisher, nor ne otherwise circulated in any form of binding or cover. Printed and bound in Athens, Greece by ATINER 8 Valaoritou Street, Kolonaki 10671 Athens, Greece www.atiner.gr ©Copyright 2015 by the Athens Institute for Education and Research. The individual essays remain the intellectual properties of the contributors. Table of Contents 1. Introduction 1 Patricia Hanna 2. Epicurean Pleasure 3 Andrew Alwood 3.
    [Show full text]
  • Foundations of Applied Mathematics I
    Foundations of Applied Mathematics I Jeffrey Ketland University of Warsaw [email protected] Forthcoming in Synthese November 23, 2020 Abstract This paper aims to study the foundations of applied mathematics, using a for- malized base theory for applied mathematics: ZFCAσ (Zermelo-Fraenkel set theory (with Choice) with atoms, where the subscript used refers to a signature specific to the application. Examples are given, illustrating the following five features of applied mathematics: comprehension principles, application conditionals, represen- tation hypotheses, transfer principles and abstract equivalents. Contents 1 Mathematicized Theories2 2 How The Laws of Set Theory Apply8 3 A Base Theory for Applied Mathematics9 4 Comprehension and Set Abstracts 17 5 Abstract Counterparts: Equivalence Modulo Mathematics 20 6 Systems and Equivalences 21 7 Application Conditionals 30 8 Representation and Transfer 34 9 Summary 37 A Four-Sorted System 38 1 1 Mathematicized Theories Standard scientific theory supplies much of the information it supplies about physical entities only indirectly, by way of apparatus pertaining to supposed relationships of physical entities to supposed mathematical entities and sup- posed classifications of and relationships among the supposed mathematical entities themselves. As much of what science says about observable entities is “theory-laden”, so much of what science says about concrete entities (observable or theoretical) is “abstraction-laden”. (Burgess & Rosen(1997): 84) 1.1 The Logical Structure of Applied Mathematics This paper aims to study the foundations of applied mathematics. Perhaps contrary to current fashions, I treat applied mathematics as taking place within a certain formalized system: specifically, Zermelo-Fraenkel set theory (with Choice) with atoms, over an application signature σ.
    [Show full text]
  • Cantorian Abstraction: a Reconstruction and Defence
    Cantorian Abstraction: A Reconstruction and Defence In section 1 of his Beitragei, Cantor explains the notion of cardinal number: We will call by the name "power" or "cardinal number" of M the general concept which, by means of our active faculty of thought, arises from the aggregate M when we make abstraction of the nature of its various elements m and of the order in which they are given. We denote the result of this double act of abstraction, the cardinal number or power of M, by (3) M (double bar). Since every single element m, if we abstract from its nature, becomes a "unit", the cardinal number M (double bar) is a definite aggregate composed of units, and this number has existence in our mind as an intellectual image or projection of the given aggregate M. Later in section 7 of the same work, he gives an analogous account of order type: Every ordered aggregate M has a definite "order type", or more shortly a definite "type", which we will denote by (2) M (bar). By this we understand the general concept which results from M if we only abstract from the nature of the elements m, and retain the order of precedence among them. Thus the order type M(bar) is itself an ordered aggregate whose elements are units which have the same order of precedence amongst one another as the corresponding elements of M, from which they are derived by abstraction. Dedekind, in §73 of his Was sind und was sollen die Zahlenii, gives a somewhat similar account of number-series: If in the consideration of a simply infinite system N set in order by a transformation _ we entirely neglect the special character of the elements, simply retaining their distinguishability and taking into account only the relations to one another in which they are placed by the order-setting transformation _, then are these elements called natural numbers or ordinal numbers or simply numbers, and the base-element 1 is called the base-number of the number-series N.
    [Show full text]
  • Gsm008-Endmatter.Pdf
    http://dx.doi.org/10.1090/gsm/008 Other Titles in This Series 8 Winfried Just and Martin Weese, Discovering modern set theory. I: The basics, 1996 7 Gerald J. Janusz, Algebraic number fields, second edition, 1996 6 Jens Carsten Jantzen, Lectures on quantum groups, 1996 5 Rick Miranda, Algebraic curves and Riemann surfaces, 1995 4 Russell A. Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock, 1994 3 William W. Adams and Philippe Loustaunau, An introduction to Grobner bases, 1994 2 Jack Graver, Brigitte Servatius, and Herman Servatius, Combinatorial rigidity, 1993 1 Ethan Akin, The general topology of dynamical systems, 1993 This page intentionally left blank Discovering Modern Set Theory. I The Basics This page intentionally left blank Graduate Studies in Mathematics Volume 8 Discovering Modern Set Theory. I The Basics Winfried Just Martin Weese Editorial Board James E. Humphreys Lance W. Small 2000 Mathematics Subject Classification. Primary 03Exx; Secondary 03E30, 03E25, 03E10, 03E05. ABSTRACT. This volume is an introduction to set theory for beginning graduate students. It covers the basics of set theory that are considered prerequisites for other areas of mathematics such as ordinals, cardinals, transfinite induction and recursion, and applications of Zorn's lemma. It also contains a description of how mathematics can be founded on axiomatic set theory, as well as a discussion of the nature of consistency results. Library of Congress Cataloging-in-Publication Data Just, W. (Winfried) Discovering modern set theory / Winfried Just, Martin Weese. p. cm. — (Graduate studies in mathematics, ISSN 1065-7339; v. 8) Includes bibliographical references and index. Contents: 1.
    [Show full text]
  • A Concise Introduction to Mathematical Logic
    Wolfgang Rautenberg A Concise Introduction to Mathematical Logic Textbook Third Edition Typeset and layout: The author Version from June 2009 corrections included Foreword by Lev Beklemishev, Moscow The field of mathematical logic—evolving around the notions of logical validity, provability, and computation—was created in the first half of the previous century by a cohort of brilliant mathematicians and philosophers such as Frege, Hilbert, Gödel, Turing, Tarski, Malcev, Gentzen, and some others. The development of this discipline is arguably among the highest achievements of science in the twentieth century: it expanded mathe- matics into a novel area of applications, subjected logical reasoning and computability to rigorous analysis, and eventually led to the creation of computers. The textbook by Professor Wolfgang Rautenberg is a well-written in- troduction to this beautiful and coherent subject. It contains classical material such as logical calculi, beginnings of model theory, and Gödel’s incompleteness theorems, as well as some topics motivated by applica- tions, such as a chapter on logic programming. The author has taken great care to make the exposition readable and concise; each section is accompanied by a good selection of exercises. A special word of praise is due for the author’s presentation of Gödel’s second incompleteness theorem, in which the author has succeeded in giving an accurate and simple proof of the derivability conditions and the provable Σ1-completeness, a technically difficult point that is usually omitted in textbooks of comparable level. This work can be recommended to all students who want to learn the foundations of mathematical logic. v Preface The third edition differs from the second mainly in that parts of the text have been elaborated upon in more detail.
    [Show full text]
  • Kalenderblätter 1001 - 1111
    Kalenderblätter 1001 - 1111 Ich übergebe sie mit zweifelhaften Gefühlen der Öffentlichkeit. Daß es dieser Arbeit in ihrer Dürftigkeit und der Finsternis dieser Zeit beschieden sein sollte, Licht in ein oder das andere Gehirn zu werfen, ist nicht unmöglich; aber freilich nicht wahrscheinlich. [Ludwig Wittgenstein: Philosophische Untersuchungen] 1001 120301 Mathematik für die ersten Semester: Top 10 1002 120302 The devil's offer 1003 120303 Enderton: Formulierungen des Auswahlaxioms 1004 120304 Heuser, Antisthenes 1005 120305 Rucker 1006 120306 Cantors Weltbild (33): Fest und unerschütterlich 1007 120307 V und Ù 1008 120308 Konstruierbar und unkonstruierbar 1009 120309 Moore 1010 120310 Cantors Weltbild (34): Verallgemeinerung des Zahlbegriffs 1011 120311 Syllogismus auf Griechisch 1012 120312 Grattan 1013 120313 Dummett 1014 120314 7. Auflage der Geschichte des Unendlichen, 3 Exemplare zu verschenken 1015 120315 Engel und große Kardinalzahlen 1016 120316 Opinions on continuum hypothesis and inaccessible cardinals 1017 120317 Schröter über Fischer 1018 120318 Fischer 1019 120319 Fischer 1020 120320 Fischer 1021 120321 Hilbert, Slater 1022 120322 Voltaire 1023 120323 Tait 1024 120324 Moore 1025 120325 Nietzsche 1026 120326 Taschner 1027 120327 Boolos 1028 120328 Kant, Galletti 1029 120329 Cantors Weltbild (35): Racenantisemitismus 1030 120330 Nietzsche über das Christentum, Cantor 1031 120331 Ramsey über Formalismus 1032 120401 In Sachen Georg Cantor 1033 120402 Corazza 1034 120403 Cattabriga 1035 120404 Mengenfolgen und Supertasks (1) 1036
    [Show full text]